Given the cutoff scale \(\Lambda \) , we evolve the running couplings from the EW scale up to \(\Lambda \)  with the RG equations.
We pick out those samples simultaneously satisfying the constraints from both vacuum stability and perturbativity, and present the results
in Fig. REF . In the figure, we show the distributions of \(\theta \) , \(\lambda _{h}\) , and \(\lambda _{s}\)  with various choices of
\(\Lambda \)  and \(\lambda _{\Lambda }\) . The constraints, of course, become stronger with a larger \(\Lambda \)  and a lower \(\lambda _{\Lambda }\) .
We find that most of the samples satisfy the vacuum stability up to a very large scale, while perturbativity can impose a stricter bound on the parameters.
For the case of \(\lambda _{\Lambda }=\lambda _{\rm FP}/2\) , the constraints with \(\Lambda =10^3\)  GeV (light green histogram) are nearly negligible
so that its distribution overlaps with the initial distribution (blue histogram). We see that in this case, most of the parameter
space of the theory remains valid for the energy scale below \(\sim 10^8\)  GeV. However, for the case of \(\lambda _{\Lambda }=\lambda _{\rm FP}/4\) ,
most of the samples are excluded for the scale above \(\sim 10^5\)  GeV. We note that the constraints from vacuum stability and perturbativity are subjective due to the somewhat arbitrary choice of the cutoff scale and the uncertainties in the value of \(\lambda _{\Lambda }\)  [1]}.
